Normalized defining polynomial
\( x^{6} - x^{5} + 59x^{4} + 65x^{3} + 341x^{2} + 337x + 1147 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-5439989503\)
\(\medspace = -\,13^{3}\cdot 19^{5}\)
|
| |
| Root discriminant: | \(41.94\) |
| |
| Galois root discriminant: | $13^{1/2}19^{5/6}\approx 41.93722377135607$ | ||
| Ramified primes: |
\(13\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-247}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_6$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(247=13\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{247}(144,·)$, $\chi_{247}(1,·)$, $\chi_{247}(246,·)$, $\chi_{247}(103,·)$, $\chi_{247}(235,·)$, $\chi_{247}(12,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-247}) \), 6.0.5439989503.1$^{3}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1549727}a^{5}-\frac{317951}{1549727}a^{4}+\frac{728845}{1549727}a^{3}+\frac{609533}{1549727}a^{2}+\frac{92976}{1549727}a-\frac{676338}{1549727}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{18}$, which has order $18$ |
| |
| Narrow class group: | $C_{18}$, which has order $18$ |
| |
| Relative class number: | $18$ |
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{8325}{1549727}a^{5}-\frac{8359}{1549727}a^{4}+\frac{453420}{1549727}a^{3}+\frac{556027}{1549727}a^{2}+\frac{711427}{1549727}a-\frac{355659}{1549727}$, $\frac{2052}{1549727}a^{5}-\frac{385}{1549727}a^{4}+\frac{103385}{1549727}a^{3}+\frac{132027}{1549727}a^{2}+\frac{170331}{1549727}a-\frac{839911}{1549727}$
|
| |
| Regulator: | \( 7.80862678603 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 7.80862678603 \cdot 18}{2\cdot\sqrt{5439989503}}\cr\approx \mathstrut & 0.236351166375 \end{aligned}\]
Galois group
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-247}) \), 3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.3.361.1 $\times$ \(\Q(\sqrt{-247}) \) $\times$ \(\Q\) |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.3.2.3a1.1 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 57 x + 121$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(19\)
| 19.1.6.5a1.4 | $x^{6} + 152$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *6 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *6 | 1.247.2t1.a.a | $1$ | $ 13 \cdot 19 $ | \(\Q(\sqrt{-247}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *6 | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.247.6t1.a.a | $1$ | $ 13 \cdot 19 $ | 6.0.5439989503.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *6 | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *6 | 1.247.6t1.a.b | $1$ | $ 13 \cdot 19 $ | 6.0.5439989503.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.